Expository notes

Mathematics

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Algebraic Topology

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See also§3.1 on the home page

a a b b a b
Fig. 1 — The torus, glued from a square. Matching edges become two loops through one point, and these generate its fundamental group.
  1. 1.1Planned

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Differential Geometry

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See also§3.2 on the home page

p v df
Fig. 2 — The differential of a function, drawn as its level sets. Its value on a vector counts, roughly, the sheets that the vector crosses.
  1. 2.1February 202657 pagesPDF, 0.7 MB

    A Motivational Introduction to Differential Forms

    An introduction to differential forms centred around intuitive understanding. Minimal prerequisites include a basic understanding of multivariable calculus and linear algebra.

  2. 2.2In progress83 pages so far

    Treatise on Differential Forms

    An exhaustive and rigorous treatment, including the homological algebra.

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Algebraic Geometry

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See also§3.3 on the home page

P Q P ∗ Q P ⊕ Q y2 = x3 − 2x + 1
Fig. 3 — Adding points on a plane cubic. The line through two points meets the curve a third time, and reflecting that point in the x-axis gives their sum.
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Analysis

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−R −ε ε R Γ γ eiz/z
Fig. 4 — The indented contour for the Dirichlet integral: a large semicircle, with a small one stepping around the pole at the origin.
  1. 4.1June 20257 pagesPDF, 0.3 MB

    Pólya Vector Fields

    Complex functions drawn as vector fields, allowing for a geometric interpretation of contour integrals and associated complex-analytic results.

  2. 4.2July 202520 pagesPDF, 0.3 MB

    The Dirichlet Integral

    The integral of sin(x)/x over the real line, proven to converge and then evaluated via various methods, including Laplace transforms and contour integration.

  3. 4.3July 202518 pagesPDF, 0.3 MB

    The Mellin Transform

    An introduction to the Mellin transform, its fundamental strip, and the Fourier and Mellin inversion theorems, used to prove Ramanujan's master theorem and, from it, Glaisher's theorem.

Discrete Mathematics

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n 0 1 2 3 4 5 n3 0 1 8 27 64 125 Δ 1 7 19 37 61 Δ2 6 12 18 24 Δ3 6 6 6 Δ4 0 0
Fig. 5 — Forward differences of the cubes. The third differences are constant, and the leading diagonal gives the cube as a combination of falling factorials.
  1. 5.1March 202510 pagesPDF, 0.3 MB

    Faulhaber's Formula and its Connection to Discrete Calculus

    A derivation of the formula for the sum of the k-th powers of the first n integers, using introductory discrete calculus: forward differences, falling factorials, and Stirling numbers of the second kind.

Probability and Statistics

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See also§3.4 on the home page

0 6 12 k Bin(12, ½)
Fig. 6 — Binomial probabilities for twelve fair coin tosses, with the normal curve of the same mean and variance.

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